Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
to switch.
For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions.
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct?
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Solve
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically.
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Using
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Return to Page Top
|
www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript B More-on-Story-Telling-and-Reason Next: [Postscript C Consistency-as-a-Tool-for-Reason.] Previous: [Postscript A Story-Telling.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28][29] [30]
Appendix B. More on
Story Telling and Reason
November 2010
Human kind has been telling stories and drawing pictures for thousands of
years. Some stories and pictures reflect reality or observations
well. Other stories and pictures may be works of imagination stemming
from dreams or done for entertainment or given for self-protection.
Stories and pictures do not have to tell the truth in order to be
followed and understood. Plays and stories may provide a virtual
reality with rules and patterns that may be ordinary or not.
The ability to follow the steps or actions in story, in sequence and in
parallel, sets the stage for the ability to give and follow instructions
and theories step by step. Dreams and nightmares recounted as
stories need not be consistent with our experience of reality. In them,
inconsistent and mutually exclusive events may be seen and imagined.
In leaving childhood, one task is to distinguish between be awake
and dreaming. However, under longterm stress or sleeplessness, we
may lose our sense of the boundary between reality and
imagination.
Rules and patterns given to describe reality may be used alone or in
combination. They may be approximate - true in some circumstances but not
all. But in chaining them together, using them directly or in
contrapositive form we may arrive at a theory of what is to tell and
check. Just a for moment, imagine that we want to compose or
extend a story that resembles reality, or what we hope reality is.
Addendum More
Once a story is partly written, we may want to extend it. In the
extension, we might have a situation A or NOT A occur. The occurrence of
the situation A is a possible addition to the story if it does not
imply an immediate contradiction or inconsistency with the earlier part
of the story. The non-occurrence of the situation A is a possible
addition to the story if it does not imply an inconsistency with an
earlier part of the story. Here the story teller could say the situation
A occurred and that would give one extension, or (s)he could say that the
situation did not occur for another extension. But then the story teller
may decide to extend the story without a mention of the occurrence or not
of the situation A. So in story telling, we have an extension in which
both A and NOT A do not occur in the story, and neither has too. So the
law of excluded middle, the statement that
A or NOT A must hold
is not true for story telling - a story composer (canonical or not)
does not have to choose A or not A. Many virtual realities are thus
possible in story telling. No connection to the five senses is required.
That makes fiction possible.
Addendum Still More
Proof by contradiction may be related to extending a story or theory by
requiring consistency.
- the situation A has to be part of a story or theory if its
non-occurrence makes the story inconsistent. Whether or not the
occurrence of A is consistent with the story or theory is another
question, and likewise,
- the situation not A (meaning the non-occurrence of A) has to be part
of the story or theory if the occurrence of A makes the story
inconsistent. Whether or not the non-occurrence of A is consistent with
the story or another is another question.
Both stories and theories may grow in tree or branch like manner, with a
shoot here, a shoot there and shoot over-there. In growing a story
or developing a theory by adding new premises or by following chains of
reason based on earlier premises and earlier chains of reason, we may not
know what is around the corner. We may not be certain the story or
its theory with all implications and premises is consistent. But we
follow the ideas and steps in a story, theory or subtheory where that is
practical, and empirically learn where, but may not know in advance if
the story and theory as is or extended is consistent in that
contradictions or mutually exclusive ideas are avoided, or is at least
consistent with reality in those cases where the story or theory fits
close enough to be useful. We may judge stories, theories and
practices (patterns and detective work included) by their internal
consistency and by their consistency with external factors. Competing
stories or theories may exist and be useful side by side.
In the physical sciences, different mechanisms to classify
ionic substances as acid, base or salt may work and agree in many
situations, but they disagree in some. So substances which should
be salts in accordance with a theory based on their chemical composition
may actually be acidic or basic (slightly and not strongly) in accordance
with observations based on litmus paper or pH-meters.
Remark: This chapter and the addendum reflect the
question of what to assume and why in mathematics and also in logic, what
methods for arriving at conclusions to assume or employ beyond the direct
use of an implication and its contrapositive. To learn more about
what might be possible, one may study mathematical logic. But in that,
the conclusion that not all is certain represents a loss of certainty,
one spoken about in a book
Mathematics: The loss of certainty, Morris Kline, Oxford
University Press, 1980-2
one that may be offset by learning about rules and patterns, their
origins, where they work and their limitations. That represent a
viable avenue in some fields of endeavour. But the loss of
certainty is a disappointment for students who study mathematics and
logic because of the apparent power and certainty of its methods in
itself and in many fields of endeavour. But that power and that
certain naively seen are finite, and so leave room for thought. Enough
said. Good luck.
Selby A, Volume 1A, Pattern Based Reason, 1996.
www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript B More-on-Story-Telling-and-Reason Next: [Postscript C Consistency-as-a-Tool-for-Reason.] Previous: [Postscript A Story-Telling.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28][29] [30]
Return to Page Top |
Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
Site Reviews
1996 - Magellan, the McKinley
Internet Directory:
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale.
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
or space.
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
cross-products.
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
trigonometry.
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why factor polynomials - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
Return to Page Top
|